Is what the inquirer did—quoting and modifying a piece of programming code, which led to the production of a similar code—considered permissible in Islamic law, especially given his intention not to sell it, but rather to use it for running his website and advertisements, and considering what was mentioned in previous fatwas that quoting and modifying are among the objectives of authorship?

If you benefit from a page to create your own code without copying it, there is no harm in that, because its owner published it for the purpose of benefiting from it, not for copying. However, copying or modifying the code without the owner's permission is an infringement on their right and effort. The agreement of ideas does not permit overstepping the rights of others in their programs and creations.